The half-life of a radioactive element is 131 days, but your sample
will not be useful to you after 90% of the radioactive nuclei
originally present have disintegrated. For about how many days can you
use the sample?

A discussion based on a calculator trick where taking 'n' repeated
square roots of 'a', then subtracting 1 and multiplying by 'b', then
adding 1 and squaring 'n' times, leads to a result very close to a^b.
The explanation is closely related to logarithms.

The Richter magnitude, R, of an earthquake is given by R=0.67
log(0.37E)+1.46 where E is the energy in kW*h released by the earthquake.
Show that if R increases by 1 unit, E increases by a factor of about 31.

A physics student wants to make sense of the various symbols used to represent "approximately equal to" -- as well as the phrase's mathematical meaning. Doctor Vogler produces two precise definitions while acknowledging that context, and personal preference, rule the day.

Doctor Carter uses logarithms and the Taylor series to show where the
72 in the "law of 72" comes from -- and shows how the same interest
rate calculations yield 115/I as an approximation for the number of
years it takes an investment that bears I interest rate to triple in
value.

I read that log scales help you see data when you are looking at
values that range largely. I also read that if the ear did not hear
logarithmically we would only hear very loud sounds. Can you help me
understand these two statements?